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Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k*(1 + x^k)/(1 - x^k)^3).
0

%I #5 Mar 23 2019 12:38:47

%S 1,1,4,15,47,160,517,1721,5668,18687,61687,203448,671253,2214377,

%T 7305308,24100319,79506903,262294336,865310405,2854666385,9417565852,

%U 31068622271,102495625503,338133855032,1115506197957,3680063534409,12140557957708,40051794232519,132131177728807

%N Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k*(1 + x^k)/(1 - x^k)^3).

%C Invert transform of A007434.

%F a(0) = 1; a(n) = Sum_{k=1..n} A007434(k)*a(n-k).

%t nmax = 28; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k] x^k (1 + x^k)/(1 - x^k)^3, {k, 1, nmax}]), {x, 0, nmax}], x]

%t a[0] = 1; a[n_] := a[n] = Sum[Sum[MoebiusMu[k/d] d^2, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 28}]

%Y Cf. A007434, A008683, A159929, A301875, A301876.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 22 2019